Optimal. Leaf size=197 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]
[Out]
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Rubi [A] time = 0.447456, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542 \[ \frac{\log \left (\sqrt{2-3 x^2}-2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\log \left (\sqrt{2-3 x^2}+2^{3/4} \sqrt [4]{2-3 x^2}+\sqrt{2}\right )}{16 \sqrt [4]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}}-\frac{\tan ^{-1}\left (\sqrt [4]{4-6 x^2}+1\right )}{8 \sqrt [4]{2}}+\frac{\tan ^{-1}\left (1-\sqrt [4]{2} \sqrt [4]{2-3 x^2}\right )}{8 \sqrt [4]{2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2-3 x^2}}{\sqrt [4]{2}}\right )}{4\ 2^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 34.2836, size = 178, normalized size = 0.9 \[ \frac{2^{\frac{3}{4}} \log{\left (- 2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{32} - \frac{2^{\frac{3}{4}} \log{\left (2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2} + \sqrt{- 3 x^{2} + 2} + \sqrt{2} \right )}}{32} - \frac{\sqrt [4]{2} \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} - 1 \right )}}{16} - \frac{2^{\frac{3}{4}} \operatorname{atan}{\left (\sqrt [4]{2} \sqrt [4]{- 3 x^{2} + 2} + 1 \right )}}{16} - \frac{\sqrt [4]{2} \operatorname{atanh}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{- 3 x^{2} + 2}}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
[Out]
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Mathematica [C] time = 0.281857, size = 139, normalized size = 0.71 \[ \frac{66 x^2 F_1\left (\frac{7}{4};\frac{3}{4},1;\frac{11}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )}{7 \left (2-3 x^2\right )^{3/4} \left (3 x^2-4\right ) \left (33 x^2 F_1\left (\frac{7}{4};\frac{3}{4},1;\frac{11}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+16 F_1\left (\frac{11}{4};\frac{3}{4},2;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )+6 F_1\left (\frac{11}{4};\frac{7}{4},1;\frac{15}{4};\frac{2}{3 x^2},\frac{4}{3 x^2}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x*(2 - 3*x^2)^(3/4)*(4 - 3*x^2)),x]
[Out]
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Maple [F] time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ( -3\,{x}^{2}+4 \right ) } \left ( -3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(-3*x^2+2)^(3/4)/(-3*x^2+4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left (3 \, x^{2} - 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247947, size = 381, normalized size = 1.93 \[ \frac{1}{128} \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (4 \, \sqrt{2} \arctan \left (\frac{2}{8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 4}}\right ) - \sqrt{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2\right ) + \sqrt{2} \log \left (8^{\frac{1}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 2\right ) + 4 \, \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 2 \, \sqrt{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 2} + 2}\right ) + 4 \, \arctan \left (\frac{2}{8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8} - 2}\right ) - \log \left (4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right ) + \log \left (-4 \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} \sqrt{-3 \, x^{2} + 2} + 8\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{3 x^{3} \left (- 3 x^{2} + 2\right )^{\frac{3}{4}} - 4 x \left (- 3 x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(-3*x**2+2)**(3/4)/(-3*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.257703, size = 284, normalized size = 1.44 \[ -\frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}} \sqrt{2} \arctan \left (-\frac{1}{8} \cdot 4^{\frac{7}{8}} \sqrt{2}{\left (4^{\frac{1}{8}} \sqrt{2} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2}{\rm ln}\left (4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) + \frac{1}{32} \cdot 4^{\frac{1}{8}} \sqrt{2}{\rm ln}\left (-4^{\frac{1}{8}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{-3 \, x^{2} + 2} + 4^{\frac{1}{4}}\right ) - \frac{1}{8} \cdot 4^{\frac{1}{8}} \arctan \left (\frac{1}{4} \cdot 4^{\frac{7}{8}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right ) - \frac{1}{16} \cdot 4^{\frac{1}{8}}{\rm ln}\left ({\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) + \frac{1}{16} \cdot 4^{\frac{1}{8}}{\rm ln}\left (-{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4^{\frac{1}{8}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/((3*x^2 - 4)*(-3*x^2 + 2)^(3/4)*x),x, algorithm="giac")
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